Monotonicity Results for Fractional Difference Operators With Discrete Exponential Kernels

Abstract

We prove that if the Caputo-Fabrizio nabla fractional difference operator ((CFR) (a-1)del(alpha) y)(t) of order 0 < alpha <= 1 and starting at a -1 is positive for t = a, a + 1,..., then y(t) is a-increasing. Conversely, if y(t) is increasing and y(a) >= 0, then ((CFR) (a-1)del(alpha)y)(t) >= 0. A monotonicity result for the Caputo-type fractional difference operator is proved as well. As an application, we prove a fractional difference version of the mean-value theorem and make a comparison to the classical discrete fractional case.